Summary:As proposed by Birch, one can construct partial Brandt matrices by the method of neighboring lattices for ternary quadratic forms.

In this talk we will present a refinement of the classical notion ofproper equivalence of lattices which leads to the construction of thefull Brandt matrices, at least in the squarefree level case. Moreover this refinement leads naturally (and is motivated by!) to thedefinition of generalized ternary theta series.

We apply these ideas to the construction of modular forms of halfintegral weight, giving an explicit version of the Shimura correspondence which generalizes results of Eichler, Gross, Ponomarev,Birch, Schulze-Pillot, and Lehman.

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